Burnley needed spot-kicks to see off League Two side Notts County in their Carabao Cup first-round tie, edging through on penalties after the sides couldn't be separated in normal time.
West Ham had a smoother night, with Jarrod Bowen and Taty Castellanos both on target as the Hammers beat Portsmouth to book their place in the next round.
Elsewhere in the competition, Norwich, Watford and Birmingham also progressed with wins, according to Sky Sports.